Constructing isoperimetric cuts

Classical isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. Such figure is a circle. Another similar problem is to find the​​ polygon of the largest possible area with a given sides. There are other isoperimetric problems related to the volume and surface area of solids of revolution.

In this paper we consider, isoperimetric cuts of various plane figures (not necessarily convex). Special cases of such cuts are the Dione problem and finding cut of the largest area from a fixed angle with the curve of given length. Solutions of this problems is that the cut have the form of a circular arc. We consider three variants of the problem: fixing the two ends of the cut, fixing one end of the cut, and without fixing the ends cut. In each case, the solution is to find the radius of the arc and the center of the circle. For any closed bounded region given problem are correct, if the length of the cutting line does not exceed the smaller diameter of the figure.

In our case considered a figure with piecewise smooth boundaries given analytically, as well as a complex figure approximated by polygons with a large number of vertices's. Depending on the shape of the figure the solution can be exact or approximate.

The main result of this paper is an algorithm for finding the maximum area of the cut (straight or curve) for a given area. The numerical experiments confirm the validity of the results.